Parameter Estimation/Tweaking Problems
solved with
Calculus Programming
_________
This section shows how to solve equations of
the following form:.
Uxx = f(x, U, Ux; a, b, c, ...)
where the parameters (a, b, c, etc.) are tweaked/estimated
This is a general form of a partial differential
equation. To tweak parameters a, b, c, etc. through Calculus-level programming we replace the following code for the "call xAxis" statement in these examples:
a = 1: b = 1: c = 1 ! Initial Estimationsfind a, b, c, ... In xAxis ooo to optimize xyz
where 'optimize' may be the term 'maximize', 'minimize', 'match',
or 'extreme'; 'match' implies your objective function
converges to zero.
plus, add an objective function in your math model.
ooo
xyz = some function of parameters a, b, c, etc.
This 'find' statement will vary your parameters until 'xyz' is optimal. Thus meeting your objective requirements.
Enjoy learning Calculus-level Programming!
Example Parameter Estimation/Tweaking Problem Source Code:
A Parameter Estimation/Tweaking Problem:
global all
problem nonLinPDE
C ------------------------------------------------------------------------
C --- Calculus Programming example: non-linear PDE (1D) Parameter
C --- Estimation Problem solved.
C ------------------------------------------------------------------------
C
C User parameters ...
! rho = ...
e0 = 8.854187817e-12 ! F/m or A2 s4 kg-1m−3 permittivity of free space
! ipoints = 10 ! grid pts. over x-axis
C
C x-parameter initial settings: x ==> i
! xFinal = 1: xPrint = xFinal/ipoints: ip = ipoints
pi= 4*atan(1)
C
a = 1: b = 1: c = 1 ! Initial Estimations
find a, b, c, ... ; in xAxis; by ?solver?; to ?optimize? xyz
end
model xAxis
C ... Integrate over x-axis
C
x= 0: xPrt = xPrint: dx = xPrt / 10
Initiate janus; for PDE;
~ equations Uxx/Ux, Ux/U; of x; step dx; to xPrt
! U = 1.23 ! @ x = 0 ... an Initial Value Problem
do while (x .lt. xFinal)
Integrate PDE; by janus
if( x .ge. xPrt) then
print 79, x, U, Ux, Uxx
xPrt = xPrt + xPrint
end if
end do
79 format( 1h , f8.4, 2x, 10(g14.5, 2x))
end
model PDE ! Partial Differential Equation
Uxx = -rho/e0 * (1.23 + sin( Ux * pi) - .543) ooo ! your Diff Eq goes here
xyz = function of parameters to be optimized ...
! e.g. xyz = U + c - b / (1 + Uxx * c**2)
end
Example Parameter Estimation/Tweaking Problem Output:
selected output goes here ...
HTML code
for linking to this page:
<a href="http://goal-driven.net/math-problems/parameter-estimation.html"><img style="margin-left: 5%; width:100px"
src="http://goal-driven.net/image/fc-compiler-icon.png"/> <strong>Example Parameter Estimation/Tweaking Problem</strong>
</a>; Simulation to Optimization, Tweak Parameters for Optimal
Solution.
